If the orbit were perfectly circular, there would still be an acceleration, and it would feel normal to us (does exist and does feel normal to us) as some decrease in our weight since it accelerates against gravity. If the earth stopped orbiting, we would be heavier (not massier of course) and maybe we would feel that, I don't know.

You are talking about a delta to this baseline always present centrifigal force that occurs because the orbit is elliptical not circular, and asking why we don't feel it. The answer must be because it is too minor a weight change for a human to notice. I have never calculated it. If it turns out to be something we should notice, I would be surpised, because your reasoning seems sound to me.

It is not like the gravitational pull of the sun, accelerates the ground or the buildings, and then the ground or the building has to accelerate us, no it doesn't work like this.

The gravitational acceleration due to the gravity of the sun is the same for each object here on earth, it is ##g_{sun}=G\frac{M_{sun}}{d^2}## where d is the distance between sun and earth..

Inside an elevator that accelerates upwards we feel acceleration because the motor accelerates the floor and then the floor pushes us upwards, but that is not how the gravity of the sun works, gravity of the sun directly pulls each and every object here on earth

No, @Grinkle - there is no acceleration experienced at all*. The two scenarios are indeed equivalent, as you can't feel any acceleration in a circular orbit, not any more than you can in an elevator whose cable snapped.

The trick here is to think about what it means to 'feel acceleration' (or a force). What does it take for a human body, or an instrument, to register acceleration, and what is special about being in free fall (or to give a more direct clue, what is special about gravity as a force).

*there's actually something to be said about tidal forces here, but we can disregard it for now, as I don't want to go on a tangent.

Earth moves around the sun in an elliptical path. When it comes close to the sun it speeds up. When it goes far, it slows down.

This is acceleration and deacceleration. Why don't we feel both of these?

I have read about it but haven't got satisfactory explanation.

The Earth speeds up and slows down in its orbit in response to the Sun's gravity, and that gravity effects all things on Earth, including us equally.

Let's use an example. Suppose you have a string of model cars attached together by rubber bands. You begin to pull on the first car to accelerate it. the rubber band between it and the second car stretches until the tension is enough to start accelerating the second car. This in turns stretches the rubber band until the third car starts to move, etc. The tension on the first rubber band has to be enough to accelerate all the cars behind it, so it stretches the most, and each band after that stretches a little less. This is like when we feel an acceleration if we push the accelerator in a car. the seat pushes on the points of your back where they are touching and this push propagates through your body, with the greatest force at the point of contact with the seat. This differential in force acting across your body is what leads to the "feeling of acceleration" you get.

Now imagine with our string of cars, that instead of just pulling the lead car, all the cars are being pulled equally by some outside force. The rubber bands do not have to transfer the motion from car to car and they all remain in their relaxed state. This is what is happening to the Earth and us when we orbit the Sun. The Earth, and all part of our bodies are responding to the gravity of the Sun and accelerating in response to it equally, so like the rubber bands in the last example, there is no differential in force acting across us to give us that feeling of acceleration.

The changes in orbital speed (over a year)would be too small to notice in any case.

If you imagine an object being pulled and pushed in a straight line at the varying Earth's orbital speed, then it would take six months to speed up and six months to slow down. This would be at a barely perceptible linear acceleration.

PS I looked it up and, of course, the Earth's orbit is nearly circular and the variation in orbital speed is only 1km/s (per 6 months).

Suppose you have a string of model cars attached together by rubber bands.

Got it! Of course - thanks. If the earth stops, we fall into the sun, we don't stay in place and get heavier. If the earth speeds up, we move farther away from the sun, being in orbit means the radial acceleration is balanced by by the centrifugal acceleration.

The changes in orbital speed (over a year)would be too small to notice in any case.

If you imagine an object being pulled and pushed in a straight line at the varying Earth's orbital speed, then it would take six months to speed up and six months to slow down. This would be at a barely perceptible linear acceleration.

PS I looked it up and, of course, the Earth's orbit is nearly circular and the variation in orbital speed is only 1km/s (per 6 months).

Though I liked your post (cause you actually looked up how much the speed varies), I believe you are missing the point: We would NOT feel the acceleration even if it was 10000Km/s(ok maybe we would feel it then cause it would have a relativistic effect :D) difference. The reason is explained by me and abit better by Janus @ post #6.

The effect of weightlessness as described in Janus' analogy depends on all elements of a body (cars) being pulled equally. That is to say, the gravitational field must be uniform - a force vector drawn from any point in the field acting on a test particle must have the same magnitude and direction.

This uniformity is approximately true far away from gravitational sources, but it is not exact. An extended object, such as e.g. a planet in orbit around a star, will experience a bit smaller force acting on its edge opposite the star than on its closer edge, and the direction towards the centre of the sun (the source of gravity) is a bit different when standing on the leading edge (or e.g. the north pole) and the trailing edge (or the south pole). Each of those differences causes a net force to emerge, stressing the body. This force is what causes tides, hence the name.
The more deviation from uniformity, the larger the effect.

However, this is not the same thing as been discussed in the question in the first post - the acceleration felt due to tidal force does not arise as a result of slowing down and accelerating in orbit.

If you could get a perfectly uniform field, and get it to change in a perfectly uniform fashion, then any body accelerated in this field would not experience any stresses - these changes in motion could not be felt.

No, gravity is unique in this respect. Speaking in Newtonian terms, it's the only force whose charge is constituted by the same quality as the one responsible for inertia - mass. As a result, acceleration in a gravitational field does not depend on the mass of the object, which leads to the observation that a multitude of massive objects accelerated by a uniform gravitational field must move in the same way, which in turn is the same as saying that you can't feel gravity acting on you (apart from the aforementioned tidal forces).

That's why we've got general relativity - the genius of Einstein had shown when he looked at the fact we're talking about, and instead of finding it somewhat amusing, like most of us would have done, he came up with the idea that gravity can be described not as a force, but as a curvature of spacetime. In GR objects moving in a gravitational field are not accelerated, they move in the space-time equivalent of straight lines (like in Newton's 1st), while gravity merely changes what it means to be a 'straight line'.